Problem 22 - Algebra 2

Question

Simplify each expression. $$ \frac{6}{a b}+\frac{8}{a} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{6 + 8b}{ab}\).

Step 1: Identify the common denominator

To add the fractions \(\frac{6}{ab}\) and \(\frac{8}{a}\), we first need to find a common denominator. The denominators are \(ab\) and \(a\). The least common denominator is \(ab\) as it is the more comprehensive term.

Step 2: Rewrite the fractions with the common denominator

The expression \(\frac{6}{ab}\) already has the common denominator \(ab\). For \(\frac{8}{a}\), we rewrite it with the denominator \(ab\) by multiplying both the numerator and the denominator by \(b\), giving us \(\frac{8b}{ab}\).

Step 3: Add the fractions

Now that both fractions have the same denominator, we can add them: \(\frac{6}{ab} + \frac{8b}{ab} = \frac{6 + 8b}{ab}\).

Step 4: Simplify the expression

The expression \(\frac{6 + 8b}{ab}\) is as simplified as possible unless there is a common factor in the numerator that can be canceled with the denominator. Since there are no common factors, this is the final simplified expression.

Key Concepts

Understanding the Common Denominator

To add fractions, we must ensure they share the same base for comparison—this base is called the 'common denominator'. It's like finding a common language between different fractions. Imagine you want to combine ingredients for a recipe, but they're measured in different units—first, you'd convert them to the same unit. Similarly, to add \( \frac{6}{ab} \) and \( \frac{8}{a} \), we identify their denominators, \( ab \) and \( a \). The smallest common term that \( ab \) and \( a \) can both fit into is \( ab \). Once we identify this common term, we can reformat each fraction to share this denominator. Establishing a common denominator is essential because it aligns the fractions, allowing for straightforward addition.

Adding Fractions with a Common Denominator

After matching denominators, the next step is quite simple—just add the numerators. For instance, we've aligned our fractions under the denominator \( ab \). Now, the fractions change to \( \frac{6}{ab} \) and \( \frac{8b}{ab} \). Think of this as stacking items in boxes of the same size. Since the sizes (denominators) match, we can proceed to combine their contents (numerators). Add the numerators: \( 6 + 8b \). Hence, the combined fraction appears as \( \frac{6 + 8b}{ab} \). Remember, it's the numerators that adjust and interact with each other, leaving the common denominator constant as a reference.

Simplifying Expressions

Let’s take a closer look at simplifying fractions. After adding the fractions, we arrive at \( \frac{6 + 8b}{ab} \). When simplifying, the aim is to reduce the expression to its simplest form. This means looking for factors common to both the numerator and the denominator that can cancel each other out. If no such factors exist, as is the case here, then the expression is already simplified. Reducing expressions makes them easier to interpret and use in further computations. Therefore, always check if there's more simplification possible, but here, with \( 6 + 8b \) and \( ab \), none can be found, making this form the final answer.